We consider the space of C-1-diffeomorphims of a three dimensional closed manifold equipped with the C-1-topology. It is known that there are open sets in which C-1-generic diffeomorphisms display uncountably many chain recurrence classes, while only countably many of them may contain periodic orbits. The classes without periodic orbits, called aperiodic classes, are the main subject of this paper. The aim of the paper is to show that aperiodic classes of C-1-generic diffeomorphisms can exhibit a variety of topological properties. More specifically, there are C-1-generic diffeomorphisms with (1) minimal expansive aperiodic classes, (2) minimal but non-uniquely ergodic aperiodic classes, (3) transitive but non-minimal aperiodic classes, (4) non-transitive, uniquely ergodic aperiodic classes.